Abstract: The Roothaan-Hartree-Fock (R-H-F) approximation in quantum mechanics plays an important role in the calculation of the electronic structure of atoms, molecules and materials, and is the basis for other high-precision electronic structure calculation methods. Based on the rich basis set resources in the open-source website Basis Set Exchange, this paper uses the characteristics of the analytic expression of the Gaussian basis set matrix elements, and the in-house Roothaan-Hartree-Fock computer program, to easily achieve high-precision calculation results. The ground state energies of various atoms and ions are calculated with basis sets of different sizes, and it is found that with the increase of the number of basis functions, different types of basis sets exhibit different convergence properties. Furthermore, using the self-written R-H-F program, this work calculated the first ionization energies of He, Be, and Ne, with a maximum error of 6.84 percent compared to the experiment.

Key words: basis set, Roothaan-Hartree-Fock equation, ground state energy, atom